Science:Math Exam Resources/Courses/MATH152/April 2016/Question B 06 (b)

MATH152 April 2016

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Question B 06 (b)

Consider the three-component differential equation ${\textbf {x}}^{\prime }=A{\textbf {x}}$. The $3\times 3$ matrix $A$ has real entries. It has an eigenvalue $\lambda _{1}=-2$ and an eigenvalue $\lambda _{2}=-1+i$ with corresponding eigenvectors ${\textbf {v}}_{1}=\left[{\begin{array}{c}1\\1\\1\end{array}}\right]$ and ${\textbf {v}}_{2}=\left[{\begin{array}{c}0\\1+i\\1\end{array}}\right].$ (b) Write the solution of the differential equation with initial data ${\textbf {x}}(0)=[1,2,3]^{T}$. Your solution must be in real form, that is is cannot involve complex numbers or complex exponentials.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint

Use the general solution found in part (a) together with the given initial data ${\textbf {x}}(0)$ to solve for the unknown constants. To obtain a solution in real form, replace the complex terms ${\textbf {v}}e^{\lambda t}$ and ${\bar {\textbf {v}}}e^{{\bar {\lambda }}t}$ that appear in the general solution by ${\text{Re}}({\textbf {v}}e^{\lambda t})$ and ${\text{Im}}({\textbf {v}}e^{\lambda t})$.

Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. From part (a), the general solution in real form is: ${\textbf {x}}(t)=C_{1}e^{-2t}\left[{\begin{array}{c}1\\1\\1\end{array}}\right]+C_{2}\left[{\begin{array}{c}0\\e^{-t}(\cos t-\sin t)\\e^{-t}\cos t\end{array}}\right]+C_{3}\left[{\begin{array}{c}0\\e^{-t}(\cos t+\sin t)\\e^{-t}\sin t\end{array}}\right].$ We are given the initial data ${\textbf {x}}(0)=\left[{\begin{array}{c}1\\2\\3\end{array}}\right].$ Recall that $e^{0}=1,\sin 0=0,$ and $\cos 0=1$. Substituting $t=0$ in the general solution, we get $\left[{\begin{array}{c}1\\2\\3\end{array}}\right]={\textbf {x}}(0)=C_{1}\left[{\begin{array}{c}1\\1\\1\end{array}}\right]+C_{2}\left[{\begin{array}{c}0\\1\\1\end{array}}\right]+C_{3}\left[{\begin{array}{c}0\\1\\0\end{array}}\right]=\left[{\begin{array}{c}C_{1}\\C_{1}+C_{2}+C_{3}\\C_{1}+C_{2}\end{array}}\right],$ which gives $C_{1}=1,C_{1}+C_{2}+C_{3}=2$ and $C_{1}+C_{2}=3.$ This implies that $C_{1}=1,C_{2}=2$ and $C_{3}=-1.$ So the solution of the differential equation satisfying the initial data is $\color {blue}{\textbf {x}}(t)=e^{-2t}\left[{\begin{array}{c}1\\1\\1\end{array}}\right]+2\left[{\begin{array}{c}0\\e^{-t}(\cos t-\sin t)\\e^{-t}\cos t\end{array}}\right]-\left[{\begin{array}{c}0\\e^{-t}(\cos t+\sin t)\\e^{-t}\sin t\end{array}}\right].$